In this paper, we propose a mathematical model of infected phytoplankton dynamics with spatial movement. The reaction diffusion models in both one and two dimension space coordinates are studied. The proposed model is an extension of temporal model available [6], in spatiotemporal domain. It is observed that the reaction diffusion system exhibits spatiotemporal chaos in phytoplankton dynamics. The importantance of the spatially extension are established in this paper, as they display a wide spectrum of ecologically relevant behavior, including chaos. The stability of the system is studied with respect to disease contact rate and the growth fraction of infected phytoplankton indirectly rejoins the susceptible phytoplankton population. The results of numerical experiments in one dimension and two dimensions in space as well as time series in temporal models are presented using MATLAB simulation. Moreover, the stability of the corresponding temporal model is studied analytically . Finally, the comparisons of the three types of numerical experimentation are discussed in conclusion. Keywords: Reaction-diffusion equation, phytoplankton dynamics, Spatiotemporal pattern formation, Chaos, Local Stability.